A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened)

Formula

T(n,k) = A000837(n/A027750(n,k)).

Example

Triangle begins:
    1
    1   1
    2   1
    3   1   1
    6   1
    7   2   1   1
   14   1
   17   3   1   1
   27   2   1
   34   6   1   1
   55   1
   63   7   3   2   1   1
  100   1
  119  14   1   1
  167   6   2   1
  209  17   3   1   1
  296   1
  347  27   7   2   1   1
  489   1
  582  34   6   3   1   1

Maple

# with table A000837 obtained from that sequence
f:= proc(n) local D, d;
  D:= sort(convert(numtheory:-divisors(n), list), `>`);
  seq(A000837[d], d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020

Mathematica

Table[Length[Select[IntegerPartitions[n], GCD@@#==k&]], {n, 20}, {k, Divisors[n]}]

Crossrefs

A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.

Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.

Sequence in context: A191861 A350200 A129761 * A207031 A156248 A324817

Adjacent sequences: A319296 A319297 A319298 * A319300 A319301 A319302

Keyword

nonn,tabf,look

Author

Gus Wiseman, Sep 16 2018

Status

approved